1/20

# 2.Fundamentals of Unconstrained Optimization英文.pdf

2.Fundamentals of Unconstrained Optimization英文.pdf

This is pag
Printer: O

C HAPTER2
Fundamentals of
Unconstrained
Optimization

In unconstrained optimization, we minimize an objective function that depends on real
variables, with no restrictions at all on the values of these variables. The mathematical
formulation is

min f (x), (2.1)
x

where x ∈ IR n is a real vector with n ≥ 1 components and f :IRn → IR is a smooth
function.
C HAPTER 2. FUNDAMENTALS OF U NCONSTRAINED O PTIMIZATION 11

y

y
3 .
.
y
2 .
y .
1 .

t

t1 t2 t3 tm

Figure 2.1 Least squares data ﬁtting problem.

Usually, we lack a global perspective on the function f . All we know are the values
of f and maybe some of its derivatives at a set of points x0, x1, x2,.... Fortunately, our
algorithms get to choose these points, and they try to do so in a way that identiﬁes a solution
reliably and without using too much computer time or storage. Often, the information
about f does not come cheaply, so we usually prefer algorithms that do not call for this
information unnecessarily.

❏ EXAMPLE 2.1

Suppose that we are trying to ﬁnd a curve that ﬁts some experimental data. Figure 2.1
plots measurements y1, y2,...,ym of a signal taken at times t1, t2,...,tm . From the data and
our knowledge of the application, we deduce that the signal has exponential and oscillatory
behavior of certain types, and we choose to model it by the function

2
−(x3−t) /x4
φ(t; x)  x1

• 页数20
• 收藏数0 收藏
• 顶次数0
• 上传人新起点
• 文件大小230 KB
• 时间2021-05-15